THE INFLUENCE OF IMPROVEMENT IN ONE MENTAL FUNCTION
UPON THE EFFICIENCY OF OTHER FUNCTIONS (I)

E. L. Thorndike & R. S. Woodworth (1901)

First published in Psychological Review, 8, 247-261.

This is the first of a number of articles reporting an inductive study
of the facts suggested by the title. It will comprise a general statement
of the results and of the methods of obtaining them, and a detailed account
of one type of experiment.

The word function is used without any rigor to refer to the mental basis
of such things as spelling, multiplication, delicacy in discrimination
of size, force of movement, marking a's on a printed page, observing
the word boy in a printed page, quickness, morality, verbal memory,
chess playing, reasoning, etc. Function is used for all sorts of qualities
in all sorts of performances from the narrowest to the widest, e.g.,
from attention to the word 'fire' pronounced in a certain tone, to attention
to all sorts of things. By the word improvement we shall mean those changes
in the workings of functions which psychologists would commonly call by
that name. Its use will be clear in each case and the psychological problem
will never be different even if the changes studied be not such as everyone
would call improvements. For all purposes 'change' may be used instead
of 'improvement' in the title. By efficiency we shall mean the status of
a function which we use when comparing individuals or the same individual
at different times, the status on which we would grade people in that function.
By other function we mean any function differing in any respect whatever
from the first. We shall at times use the word function-group to mean [p.
248] those cases where most psychologists would say that the same operation
occurred with different data. The function attention, for instance,
is really a vast group of functions.

Our chief method was to test the efficiency of some function or functions,
then to give training in some other function or functions until a certain
amount of improvement was reached, and then to test the first function
or set of functions. Provided no other factors were allowed to affect the
tests, the difference between the test before and the test after training
measures the influence of the improvement in the trained functions on the
functions tested.

It is possible to test the general question in a much neater and more
convenient way by using, instead of measures of a function before and after
training with another, measures of the correlation between the two functions.
If improvement in one function increases the efficiency of another and
there has been improvement in one, the other should be correlated with
it; the individuals who have high rank in the one should have a higher
rank in the other than the general average. Such a result might also be
brought about by a correlation of the inborn capacities for those functions.
Finding correlation between two functions thus need not mean that improvement
in one has brought increased efficiency in the other. But the absence of
correlation does mean the opposite. In an unpublished paper Mr. Clark Wissler,
of Columbia University, demonstrates the absence of any considerable correlation
between the functions measured by the tests given to students there. Miss
Naomi Norsworthy, of Teachers College, has shown (the data were presented
at the Baltimore meeting; the research is not yet in print) that there
is no correlation between accuracy in noticing misspelled words and accuracy
in multiplication, nor between the speeds; that there is little or no correlation
between accuracy and speed in marking on a printed page misspelled words,
words containing r and e, the word boy, and in marking
semi-circles on a page of different geometrical figures.

Perhaps the most striking method of showing the influence or lack of
influence of one function on another is that of testing the same function-group,
using cases where there are very [p. 249] slightly different data. If,
for instance, we test a person's ability to estimate a series of magnitudes
differing each from the next very slightly, and find that he estimates
one very much more accurately than its neighbors on either side, we can
be sure that what he has acquired from his previous experience or from
the experience of the test is not improvement in the function-group of
estimating magnitudes but a lot of particular improvements in estimating
particular magnitudes, improvements which may be to a large extent independent
of each other.

The experiments, finally, were all on the influence of the training
on efficiency, on ability as measured by a single test, not on the ability
to improve. It might be that improvement in one function might fail
to give in another improved ability, but succeed in giving ability to improve
faster than would have occurred had the training been lacking.

The evidence given by our experiments makes the following conclusions
seem probable.

It is misleading to speak of sense discrimination, attention, memory,
observation, accuracy, quickness, etc., as multitudinous separate individual
functions are referred to by any one of these words. These functions may
have little in common. There is no reason to suppose that any general change
occurs corresponding to the words 'improvement of the attention,' or 'of
the power of observation,' or 'of accuracy.'

It is even misleading to speak of these functions as exercised within
narrow fields as units. For example, 'attention to words' or 'accurate
discrimination of lengths' or 'observation of animals' or 'quickness of
visual perception' are mythological, not real entities. The words do not
mean any existing fact with anything like the necessary precision for either
theoretical or practical purposes, for, to take a sample case, attention
to the meaning of words does not imply equal attention to their spelling,
nor attention to their spelling equal attention to their length, nor attention
to certain letters in them equal attention to other letters.

The mind is, on the contrary, on its dynamic side a machine for making
particular reactions to particular situations. It works in great detail,
adapting itself to the special data of which [p. 250] it has had experience.
The word attention, for example, can properly mean only the sum
total of a lot of particular tendencies to attend to particular sorts of
data, and ability to attend can properly mean only the sum total of all
the particular abilities and inabilities, each of which may have an efficiency
largely irrespective of the efficiencies of the rest.

Improvement in any single mental function need not improve the ability
in functions commonly called by the same name. It may injure it.

Improvement in any single mental function rarely brings about equal
involvement in any other function, no matter how similar, for the working
of every mental function-group is conditioned by the nature of the data
in each particular case.

The very slight amount of variation in the nature of the data necessary
to affect the efficiency of a function-group makes it fair to infer that
no change in the data, however slight, is without effect on the function.
The loss in the efficiency of a function trained with certain data, as
we pass to data more and more unlike the first, makes it fair to infer
that there is always a point where the loss is complete, a point beyond
which the influence of the training has not extended. The rapidity of this
loss, that is, its amount in the case of data very similar to the data
on which the function was trained, makes it fair to infer that this point
is nearer than has been supposed.

The general consideration of the cases of retention or of loss of practice
effect seems to make it likely that spread of practice occurs only where
identical elements are concerned in the influencing and influenced function.

The particular samples of the influence of training in one function
on the efficiency of other functions chosen for investigation were as follows:

1. The influence of certain special training in the estimation of magnitudes
on the ability to estimate magnitudes of the same general sort, i.
e., lengths or areas or weights, differing in amount, in accessory
qualities (such as shape, color, form) or in both. The general method was
here to test the subject's accuracy of estimating certain magnitudes, e.
g., lengths of lines. He would, that is, guess the length of each.
Then he would [p. 251] practice estimating lengths within certain limits
until he attained a high degree of proficiency. Then he would once more
estimate the lengths of the preliminary test series. Similarly with weights,
areas, etc. This is apparently the sort of thing that happens in the case
of a tea-tester, tobacco-buyer, wheat-taster or carpenter, who attains
high proficiency in judging magnitudes or, as we ambiguously say, in delicacy
of discriminating certain sense data. It is thus like common cases of sense
training in actual life.

2. The influence of training in observing words containing certain combinations
of letters (e.g., s and e) or some other characteristic
on the general ability to observe words. The general method here was to
test the subject's speed and accuracy in picking out and marking certain
letters, words containing certain letters, words of a certain length, geometric
figures, misspelled words, etc. He then practiced picking out and marking
words of some one special sort until he attained a high degree of proficiency.
He was then re-tested. The training here corresponds to a fair degree with
the training one has in learning to spell, to notice forms and endings
in studying foreign languages, or in fact in learning to attend to any
small details.

3. The influence of special training in memorizing on the general ability
to memorize. Careful tests of one individual and a group test of students
confirmed Professor James' result (see Principles of Psychology, Vol. I.,
pp. 666-668). These tests will not be described in detail.

These samples were chosen because of their character as representative
mental functions, because of their adaptability to quantitative interpretations
and partly because of their convenience. Such work can be done at odd times
without any bulky or delicate apparatus. This rendered it possible to secure
subjects. In all the experiments to be described we tested the influence
of improvement in a function on other functions closely allied to it.
We did not in sense-training measure the influence of training one sense
on others, nor in the case of training of the attention the influence of
training in noticing words on, say, the ability to do mental arithmetic
or to listen to a metaphysical discourse. For common observation seemed
to give a negative [p. 252] answer to this question, and some considerable
preliminary experimentation by one of us supposed such a negative. Mr.
Wissler's and Miss Norsworthy's studies are apparently conclusive, and
we therefore restricted ourselves to the more profitable inquiry.

A SAMPLE EXPERIMENT.

There was a series of about 125 pieces of paper cut in various shapes.
(Area test series.) Of these 13 were rectangles of almost the same shape
and of sizes from 20 to 90 sq. cm. (series 1), 27 others were triangles,
circles, irregular figures, etc., within the same limits of size (series
2). A subject was given the whole series of areas and asked to write down
the area in sq. cm. of each one. In front of him was a card on which three
squares, 1, 25 and 100 sq. cm. in area, respectively, were drawn. He was
allowed to look at them as much as he pleased but not to superpose the
pieces of paper on them. No other means of telling the areas were present.
After being thus tested the subject was given a series of paper rectangles,[1]
from 10 to 100 sq. cm. in area and of the same shape as those of series
1. These were shuffled and the subject guessed the area of one, then looked
to see what it really was and recorded his error. This was continued and
the pieces of paper were kept shuffled so that he could judge their area
only from their intrinsic qualities. After a certain amount of improvement
had been made he was re-tested with the 'area test series' in the same
manner as before.

[p. 253] The function trained was that of estimating areas from 10 to
100 sq. cm. with the aid of the correction of wrong tendencies supplied
by ascertaining the real area after each judgment. We will call this 'function
a.' A certain improvement was noted. What changes in the efficiency
of closely allied functions are brought about by this improvement? Does
the improvement in this function cause equal improvement (1) in the function
of estimating areas of similar size but different shape without the correction
factor? or (2) in the function of estimating identical areas without the
correction factor? (3) In any case how much improvement was there? (4)
Is there as much improvement in the function of estimating dissimilar shapes
as similar? The last is the most important question.

We get the answer to 1 and part of 3 by comparing in various ways the
average errors of the test areas of dissimilar shape in the before and
after tests. These are given in Table I. The average errors for the last
trial of the areas in the training series similar in size to the test series
are given in the same table.

The function of estimating series 2 (same sizes, different shapes) failed
evidently to reach an efficiency equal to that of the function trained.
Did it improve proportionately as much?

This is a hard question to answer exactly, since the efficiency or 'function
a' increases with great rapidity during the first score or so of
trials, so that the average error of even the first twenty estimates made
is below that of the first ten, and that again is below that of the first
five. Its efficiency at the start depends thus on what you take to be the
start. The fact is that the first estimate of the training series is not
an exercise of 'function a' at all and that the correction
influence increases [p. 254] up to a certain point which we cannot exactly
locate. The fairest method would seem to be to measure the improvement
in 'function a' from this point and compare with that improvement
the improvement in the other function or functions in question. This point
is probably earlier in the series than would be supposed. If found, it
would probably make the improvement in 'function a' greater than
that given in our percentages.

The proportion of average error in the after test to that in the before
test is greater in the case of the test series than in the case of the
first and last estimations of the areas of the same size in the training
series, save in the case of Be. The proportions are given in the
following table:

Question 2 is answered by a comparison of the average errors, before and
after the training, of Series 1. (identical areas) given without the correction
factor. The efficiency reached in estimating without the correction factor
(see column 2 of Table III.) is evidently below that reached in 'function
a.' The results there in the case of the same areas are given in
column 3.

[p. 255] The function of estimating an area while in the frame of mind
due to being engaged in estimating a limited series of areas and seeing
the extent of one's error each time, is evidently independent to a large
extent of the function of judging them after the fashion of the tests.

If we ask whether the function of judging without correction improved
proportionately as much as 'function a,' we have our previous difficulty
about finding a starting point for a. Comparing as before the first
100 estimates with the last 100 we get the proportions in the case of the
areas identical with those in the test. These are given in column 7. The
proportions in the case of the test areas (series 1; same shape) are given
in column 6. A comparison of columns 6 and 7 thus gives more or less of
an answer to the question, and column 6 gives the answer to the further
one: "How much improvement was there?"

We can answer question 4 definitely. Column 5 repeats the statement
of the improvement in the case of the test areas of different shape, and
by comparing column 6 with it we see that in every case save that of Be.
there was more improvement when the areas were similar in shape to those
of the training series. This was of course the most important fact to be
gotten at.

To sum up the results of this experiment, it has been shown that the
improvement in the estimation of rectangles of a certain shape is not equalled
in the case of similar estimations of areas of different shapes. The function
of estimating areas is really a function-group, varying according to the
data (shape, size, etc.). It has also been shown that even after mental
standards of certain limited areas have been acquired, the function of
estimating with these standards constantly kept alive by noticing the real
area after each judgment is a function largely independent of the function
of estimating them with the standards fully acquired by one to two thousand
trials, but not constantly renewed by so noticing the real areas.
Just what happened in the training was the partial formation of a number
of associations. These associations were between sense impressions of particular
sorts in a particular environment coming to a person in a particular mental
attitude or frame of mind, and a number of ideas or impulses.

[p. 256] What was there in this to influence other functions, other
processes than these particular ones? There was first of all the acquisition
of certain improvements in mental standards of areas. These are of some
influence in judgments of different shapes. We think, "This triangle or
circle or trapezoid is about as big as such and such a rectangle, and such
a rectangle would be 49 sq. cm." The influence is here by means of an idea
that may form an identical element in both functions. Again, we may form
a particular habit of making a discount for a tendency to a constant error
discovered in the training series. We may say, "I tend to judge with a
minus error," and the habit of thinking of this may be beneficial in all
cases. The habit of bearing this judgment in mind or of unconsciously making
an addition to our first impulse is thus an identical element of both functions.
This was the case with Be. That there was no influence due to a
mysterious transfer of practice, to an unanalyzable property of mental
functions, is evidenced by the total lack of improvement in the functions
tested in the case of some individuals.

On pushing our conception of the separateness of different functions
to its extreme, we were led to ask if the function of estimating one magnitude
might not be independent even of the functions of estimating magnitudes
differing only slightly from the first. It might be that even the judgment
of areas of 40-50 sq. cm. was not a single function, but a group of similar
functions, and that ability might be gained in estimating one of these
areas without spreading to the others. The only limits that must necessarily
be set to this subdivision would be those of the mere sensing of small
differences.

If, on the contrary, judgments of nearly equal magnitudes are acts of
a single function, ability gained in one should appear in the others also.
The results of training should diffuse readily throughout the space covered
by the function in question, and the accuracy found in judgments of different
magnitudes within this space should be nearly constant. The differences
found should simply be such as would be expected from chance.

The question can be put to test by comparing the actual difference between
the average errors made, in judging each of [p. 257] neighboring magnitudes,
with the probable difference as computed from the probability curve. If
the actual difference greatly exceeds the probable difference, it is probably
significant of some real difference in the subject's ability to judge the
two magnitudes. He has somehow mastered one better than the other. No matter
how this has come about. If it is a fact, then clearly ability in the one
has not been transferred to the other.

Our experiments afford us a large mass of material for testing this
question. In the 'training series,' we have a considerable number (10 to
40) of judgments of each of a lot of magnitudes differing from each other
by slight amounts. We have computed the accuracy of the judgment of each
magnitude (as measured by the error of mean square), and then compared
the accuracy for each with that for the adjacent magnitudes. We find many
instances in which the difference between the errors for adjacent magnitudes
is largely in excess of the probable difference. And the number of such
instances greatly exceeds what can be expected from chance. [2]

These great differences between the errors of adjacent magnitudes are
strikingly seen in the curves on page 259. The ordinates of these curves
represent the mean square error of judgments of areas of 10 to 100 square
centimeters, and for 3 individuals. The dots above and below each point
of the curve give the 'limits of error' of that value, as determined by
the formula,, in which m
is the error of mean square, and n the number of cases. These limits
are such that the odds are about 2 to 1, more exactly 683 to 317, that
the true value lies inside them. The dots thus furnish a measure of the
reliability of the curve at every point.

[p. 258] These curves are all irregular, with sudden risings and fallings
that greatly obscure their general course. Psychologists are familiar of
old with irregularities of this kind, and are wont to regard them as effects
of chance, and so to smooth out the curve. But as we find more irregularity
than can reasonably be attributed to chance, we conclude that our curves
at least should not be smoothed out, and that the sudden jumps, or some
of them, signify real differences in the person's ability.[3]

If, for example, we examine Fig. 1, we notice a number of sudden jumps,
or points at which the errors in judging adjacent magnitudes differed considerably
from each other. The most significant of these jumps are at 10-11, 36-37,
41-42, 65-66, 66-67, 83-84, and 98-99 sq. cm. The question is whether such
a jump as that at 41-42 indicates greater ability to judge 42 sq. cm.,
or whether the observed difference is simply due to chance and the relatively
few cases (here 10 for each area). A vague appeal to chance should not
be allowed, in view of the possibility of calculating the odds in favor
of each side of the question. This can be done by a fairly simple method.
We can consider two adjacent areas as practically equal, so far as concerns
Weber's law or any similar law. The average errors found for the two would
thus be practically two determinations of the same quantity, and should
differ only as two determinations of the same quantity may probably differ.

We wish then to compare the actual difference between the errors for
41 and 42 sq. cm. with the probable difference. The error -- we use throughout
the 'error of mean square,' and the measure of reliability based on it
-- this error is here 6.2 and 3.1 sq. cm. respectively. The actual difference
is 3.1 sq. cm. To find the probable difference, we first find the 'limits
of error' or reliability of each determination, as described above, and
then find the square root of the sums of the squares of these

[p. 259]

[p. 260] 'limits of error.' The 'limits' are here 10 and 0.7, and the probable
difference 1.2 sq. cm. The actual difference is 2.6 times the probable.
In this whole series we find 6 other instances in which the actual difference
is over 2 times the probable. From the probability integral we find that,
in the long run, 46 actual differences to the thousand would exceed twice
the probable. The question is, therefore, what is the probability of finding
as many as 7 such differences in a series of 90? This is a form of the
familiar problem in probabilities: to find the chances that an event whose
probability is p shall occur at least r times out of a possible
n. The solution depends on an application of the binomial theorem,
and may be evaluated by means of logarithms. In the present case, the value
found is .1209 or about 1/8.

Instead of vaguely saying that the large jumps seen in the curves may
be due to chance, we are now able to state that the odds are 7 to 1 against
this view, and 7 to 1 in favor of the view that the large jumps, or some
of them, are significant of inequality in the person's power to estimate
nearly equal areas. These odds are of course not very heavy form the standpoint
of scientific criticism. But they are fortified by finding, as we do, the
same general balance of probability in all of the series examined. In one
other series, the number of large differences is small, and the probability
is as large as .2938 that they are due to mere chance. In three other series,
this probability is very small, measuring .0025, .0025, .0028, or about
1/400. Finally, in the series corresponding to Fig. 3, there are a large
number of actual differences which far exceed the probable. (The errors
are small, and consequently the probable differences are small.) There
are 31 that exceed twice the probable difference, and of these 9 exceed
3.5 times the probable difference. The probability of finding even these
9 is so small that six-place logarithms cannot determine it exactly, but
it is less than .000001.

In four cases, then, out of six examined, it is altogether inadmissible
to attribute the differences to chance, while in the other two the odds
are against doing so. The probability that the differences in all
the series are due to chance is of course [p. 261] multiply small. The
differences are therefore not chance, but significant; the ability to judge
one magnitude is sometimes demonstrably better than the ability to judge
the next magnitude; one function is better developed than its neighbor.
The functions of judging nearly equal magnitudes are, sometimes at least,
largely separate and independent. A high degree of ability in one sometimes
coexists with a low degree of ability in the others.

Footnotes

[1] The judgments of area were made with the following
apparatus: a series of parallelograms ranging from 10 to 140 and from 190
to 280 sq. cm., varying each from the next by 1 sq. cm. Their proportions
were almost the same (no one of them could possibly be distinguished by
its shape). For example, the dimensions of those from 137 to 145 sq. cm.
were

[2] The smaller error at certain magnitudes is not
the result of a preference of the subject to guess that number. Of course,
if the subject were prone to guess '64 square centimeters' oftener than
63 or 65, he would be more apt to guess 64 right, and the error for 64
would be diminished. We therefore made a few tables of the frequency with
which each number was guessed. But we found that the magnitudes that were
best judged were not more often guessed than their neighbors.

[3] The fact that judgments of nearly equal magnitudes
may show very unequal errors throws doubt on all curves drawn from the
judgment of only a few 'normals.' If slightly different normals had been
chosen, the errors might have been considerably different, and the course
of the curve changed. If, for example, three normals be chosen from the
91 in our curves, and those three used as the basis of a curve, the curve
will vary widely with the choice of the three normals.